86 M. M. SLOTNICK 
If the curvatures & of the plane normal sections be plotted against 
d for o° SA< 180°, the curve would have the characteristic form shown 
in Figure 3. If the curvature changes sign, as A varies, the correspond- 
ing curve crosses the \-axis twice. 
It has been shown that the difference in the values of \ for which 
k is a maximum and that for which it is a minimum is go°. Moreover, 
the algebraic sum of the ordinates (&) for any two values of \ whose 
difference is go° is a constant. 
The directions on the surface at P in which these extreme values 
of the normal curvature occur are called the principal directions. The 
reciprocals, p; and pe, of the extreme values of the curvatures, k; and 
ke, are called the radii of principal curvatures: 
I I 
ki=— k=—- 
Pl P2 
The total or Gaussian curvature of the surface at the point is defined 
as the product hike; and one-half the sum: 3 (&i+2) is the mean 
curvature of the surface at the point. 
Many further interesting properties of the curvature of a surface 
can readily be deduced from the analytical method here developed. 
However, it is not necessary for our purpose to delve into these. It is 
well to point out before leaving the matter, that, though, initially, 
we chose the x and y axes as any two mutually perpendicular lines 
in the tangent plane to S at P, we may now, in the light of our re- 
sults, choose them in the two principal directions. Analytically, this 
means that the solutions of (16) are \=o° and A=go°; that is, s=o 
in the series expansion (3). 
The equation (16) leads to the results: 
ae) 2S + (r — 2) 
ea eet ae eae 
= » COS 2A = 
w w 

(17) sin 2\ = 
where 
(18) w= VJ(r—d? + 4s?. 
The positive signs in (17) yield the value of \ for which the curvature 
attains one extreme value, and the negative signs yield the other 
value of A, differing from the first by 90°, for which the curvature 
attains the other extreme value. If we set either one of these-values 
of \ for A; in (12) and (13), the &; and kz so obtained will be the values 
of these extreme curvatures. 
416 
